What is the relationship between degrees of freedom and the size of the training dataset?
When you define a straight line of the form $y=mx+c$, you need 2 points $(x_1,y_1)$ and $(x_2,y_2)$, to solve for the 2 variables $m$ and $c$ (you can easily visualise this graphically). Similarly, a parabola of the form $y=ax^2+bx+c$ will require 3 such points.

Now viewing it as a ML problem, you are given the points and you have to estimate the parameters such that the training error is 0 (Regression). So just like the previous case you have a bunch of $(x_i,y_i)$ and you have to fit a curve whose degree of freedom you have to choose. Here $m,c,a,b$ are all replaced with more generic $w$ called as a parameter

If you have $10$ degree of freedom and $10$ data-points you can solve for the parameters of the model (unambiguous solution i.e only one and one unique solution will exist). Whereas , if the degree of freedom is lower you'll get a solution which may miss one point. For, example if you are given 3 points and ask to fit a straight line through it, you may or may not be able to (depending on collinearity). On the opposite case if you have more degrees of freedom you can get multiple values for a single parameter. Assuming a single extra degree of freedom, you can take a parameter hold it to some fixed value and solve the rest of the equation as mentioned above you get some value for the rest of parameters. Now hold the same parameter at a different fixed value and repeat the same process, you get some different value of the other parameters.

In general, it is easier to view it this way:

If let's say you have 3 degrees of freedom $y=w_2x_1^2+w_1x_1+w_0$ and 3 data-points $(x_1,y_1), (x_2,y_2), (x_3,y_3)$, you can get a system of equation:

$$y_1=w_2x_1^2+w_1x_1+w_0$$
$$y_2=w_2x_2^2+w_1x_2+w_0$$
$$y_3=w_2x_3^2+w_1x_3+w_2$$

Thus you get a system of linear equations (NOTE: $x_i,y_i$ are known here). The only unknowns here are $w_i$'s for which there are multiple methods to solve. You can extend this for $n$ equations.